Dynamic Foraging Patterns in Antarctic Humpbacks
Ben Weinstein
2017-05-21 20:47:42
## Source: local data frame [2 x 2]
##
## Animal max(timestamp, na.rm = T)
## (int) (time)
## 1 123232 2013-09-28 07:28:00
## 2 123236 2013-03-18 11:26:00
0.1 By Month
1 Correlated random walk
Process Model
\[ d_{t} \sim T*d_{t-1} + Normal(0,\Sigma)\] \[ x_t = x_{t-1} + d_{t} \]
1.1 Parameters
For each individual:
\[\theta = \text{Mean turning angle}\] \[\gamma = \text{Move persistence} \]
For both behaviors process variance is: \[ \sigma_{latitude} = 0.1\] \[ \sigma_{longitude} = 0.1\]
1.2 Behavioral States
\[ \text{For each individual i}\] \[ Behavior_1 = \text{traveling}\] \[ Behavior_2 = \text{foraging}\]
\[ \alpha_{i,1,1} = \text{Probability of remaining traveling when traveling}\] \[\alpha_{i,2,1} = \text{Probability of switching from Foraging to traveling}\]
\[\begin{matrix} \alpha_{i,1,1} & 1-\alpha_{i,1,1} \\ \alpha_{i,2,1} & 1-\alpha_{i,2,1} \\ \end{matrix}\]
With the probability of switching states:
\[logit(\phi_{traveling}) = \alpha_{Behavior_{t-1}}\]
\[\phi_{foraging} = 1 - \phi_{traveling} \]
1.3 Continious tracks
The transmitter will often go dark for 10 to 12 hours, due to weather, right in the middle of an otherwise good track. The model requires regular intervals to estimate the turning angles and temporal autocorrelation. As a track hits one of these walls, call it the end of a track, and begin a new track once the weather improves. We can remove any micro-tracks that are less than three days. Specify a duration, calculate the number of tracks and the number of removed points. Iteratively.
1.3.1 After filitering
How did the filter change the extent of tracks?
sink(“Bayesian/Multi_RW.jags”) cat(" model{
#Constants
pi <- 3.141592653589
#for each if 6 argos class observation error
for(x in 1:6){
##argos observation error##
argos_prec[x,1:2,1:2] <- argos_cov[x,,]
#Constructing the covariance matrix
argos_cov[x,1,1] <- argos_sigma[x]
argos_cov[x,1,2] <- 0
argos_cov[x,2,1] <- 0
argos_cov[x,2,2] <- argos_alpha[x]
}
for(i in 1:ind){
for(g in 1:tracks[i]){
## Priors for first true location
#for lat long
y[i,g,1,1:2] ~ dmnorm(argos[i,g,1,1,1:2],argos_prec[1,1:2,1:2])
#First movement - random walk.
y[i,g,2,1:2] ~ dmnorm(y[i,g,1,1:2],iSigma)
###First Behavioral State###
state[i,g,1] ~ dcat(lambda[]) ## assign state for first obs
#Process Model for movement
for(t in 2:(steps[i,g]-1)){
#Behavioral State at time T
phi[i,g,t,1] <- alpha[state[i,g,t-1],Month[i,g,t]]
phi[i,g,t,2] <- 1-phi[i,g,t,1]
state[i,g,t] ~ dcat(phi[i,g,t,])
#Turning covariate
#Transition Matrix for turning angles
T[i,g,t,1,1] <- cos(theta[state[i,g,t]])
T[i,g,t,1,2] <- (-sin(theta[state[i,g,t]]))
T[i,g,t,2,1] <- sin(theta[state[i,g,t]])
T[i,g,t,2,2] <- cos(theta[state[i,g,t]])
#Correlation in movement change
d[i,g,t,1:2] <- y[i,g,t,] + gamma[state[i,g,t],Month[i,g,t]] * T[i,g,t,,] %*% (y[i,g,t,1:2] - y[i,g,t-1,1:2])
#Gaussian Displacement
y[i,g,t+1,1:2] ~ dmnorm(d[i,g,t,1:2],iSigma)
}
#Final behavior state
phi[i,g,steps[i,g],1] <- alpha[state[i,g,steps[i,g]-1],Month[i,g,steps[i,g]-1]]
phi[i,g,steps[i,g],2] <- 1-phi[i,g,steps[i,g],1]
state[i,g,steps[i,g]] ~ dcat(phi[i,g,steps[i,g],])
## Measurement equation - irregular observations
# loops over regular time intervals (t)
for(t in 2:steps[i,g]){
# loops over observed locations within interval t
for(u in 1:idx[i,g,t]){
zhat[i,g,t,u,1:2] <- (1-j[i,g,t,u]) * y[i,g,t-1,1:2] + j[i,g,t,u] * y[i,g,t,1:2]
#for each lat and long
#argos error
argos[i,g,t,u,1:2] ~ dmnorm(zhat[i,g,t,u,1:2],argos_prec[argos_class[i,g,t,u],1:2,1:2])
}
}
}
}
###Priors###
#Process Variance
iSigma ~ dwish(R,2)
Sigma <- inverse(iSigma)
##Mean Angle
tmp[1] ~ dbeta(10, 10)
tmp[2] ~ dbeta(10, 10)
# prior for theta in 'traveling state'
theta[1] <- (2 * tmp[1] - 1) * pi
# prior for theta in 'foraging state'
theta[2] <- (tmp[2] * pi * 2)
##Move persistance
# prior for gamma (autocorrelation parameter) in state 1
#for each month
for (m in 1:Months){
#Intercepts
logit(alpha[1,m]) <- alpha_mu_invlogit[1,m]
alpha_mu_invlogit[1,m] ~ dnorm(alpha_mu[1],alpha_tau[1])
logit(alpha[2,m]) <- alpha_mu_invlogit[2,m]
alpha_mu_invlogit[2,m] ~ dnorm(alpha_mu[2],alpha_tau[2])
gamma[1,m] ~ dbeta(3,2) ## gamma for state 1
dev[m] ~ dbeta(1,1) ## a random deviate to ensure that gamma[1] > gamma[2]
gamma[2,m] <- gamma[1,m] * dev[m]
}
##Behavioral States
#Hierarchical structure across months
#Switching among states in inv.logit space
alpha_mu[1] ~ dnorm(0,0.386)
alpha_mu[2] ~ dnorm(0,0.386)
#Variance in state change per month
alpha_tau[1] ~ dgamma(0.0001,0.0001)
alpha_tau[2] ~ dgamma(0.0001,0.0001)
#Probability of behavior switching
lambda[1] ~ dbeta(1,1)
lambda[2] <- 1 - lambda[1]
##Argos priors##
#longitudinal argos precision, from Jonsen 2005, 2016, represented as precision not sd
#by argos class
argos_sigma[1] <- 11.9016
argos_sigma[2] <- 10.2775
argos_sigma[3] <- 1.228984
argos_sigma[4] <- 2.162593
argos_sigma[5] <- 3.885832
argos_sigma[6] <- 0.0565539
#latitidunal argos precision, from Jonsen 2005, 2016
argos_alpha[1] <- 67.12537
argos_alpha[2] <- 14.73474
argos_alpha[3] <- 4.718973
argos_alpha[4] <- 0.3872023
argos_alpha[5] <- 3.836444
argos_alpha[6] <- 0.1081156
}"
,fill=TRUE)sink()
## user system elapsed
## 4.577 0.608 10677.5561.4 Chains
## used (Mb) gc trigger (Mb) max used (Mb)
## Ncells 1316965 70.4 2637877 140.9 2637877 140.9
## Vcells 67221447 512.9 154967408 1182.4 142613440 1088.1## used (Mb) gc trigger (Mb) max used (Mb)
## Ncells 1309557 70.0 2637877 140.9 2637877 140.9
## Vcells 23009580 175.6 123973926 945.9 142613440 1088.1
1.5 Change in autocorrelation over time
2 Change in transition probabilities over time
2.1 Parameter Summary
## parameter par mean lower upper
## 1 alpha alpha[1,1] 0.86889058 0.70736008 0.9599329
## 2 alpha alpha[2,1] 0.07749341 0.02926343 0.1607627
## 3 alpha alpha[1,2] 0.89731036 0.81892518 0.9629942
## 4 alpha alpha[2,2] 0.07080047 0.02942902 0.1324746
## 5 alpha alpha[1,3] 0.87551786 0.74335947 0.9595886
## 6 alpha alpha[2,3] 0.06220564 0.02080083 0.1173844
## 7 alpha alpha[1,4] 0.83635962 0.48025207 0.9593019
## 8 alpha alpha[2,4] 0.07515583 0.02464966 0.1581212
## 9 alpha_mu alpha_mu[1] 1.99076996 0.82174095 2.9685912
## 10 alpha_mu alpha_mu[2] -2.62749126 -3.42910697 -1.7245048
## 11 alpha_tau alpha_tau[1] 956.22087995 0.22199589 5501.2365980
## 12 alpha_tau alpha_tau[2] 1067.43994161 0.48344033 5700.4313655
## 13 gamma gamma[1,1] 0.52928441 0.25709448 0.7981171
## 14 gamma gamma[2,1] 0.23499121 0.02237729 0.5128794
## 15 gamma gamma[1,2] 0.72591753 0.59979522 0.8489317
## 16 gamma gamma[2,2] 0.19800407 0.01407748 0.4848036
## 17 gamma gamma[1,3] 0.61199329 0.41289264 0.8235036
## 18 gamma gamma[2,3] 0.29428258 0.06754258 0.5242353
## 19 gamma gamma[1,4] 0.45170657 0.17063885 0.7845834
## 20 gamma gamma[2,4] 0.18017117 0.01547798 0.4316955
## 21 theta theta[1] 0.06636237 -0.03920256 0.1762299
## 22 theta theta[2] 3.21355391 2.69634661 3.7480006
3 Behavioral Prediction
3.1 Spatial Prediction
3.2 By individual
3.3 Autocorrelation in behavior
3.3.1 As single timeline
4 Simulated tracks
4.1 Behavioral description
4.1.1 Proportion of states by month
4.1.2 Distance between bouts
5 Behavior duration
## Source: local data frame [12 x 7]
## Groups: Animal, Track, Bout, phistate [12]
##
## Animal Track Bout phistate MonthF Days Month
## (dbl) (dbl) (int) (fctr) (dbl) (dbl) (fctr)
## 1 1 1 2 Traveling 2 1.3513889 February
## 2 1 1 4 Traveling 3 2.4715278 March
## 3 1 1 6 Traveling 4 1.4750000 April
## 4 2 1 1 Traveling 2 6.3402778 February
## 5 2 1 3 Traveling 2 2.8000000 February
## 6 2 1 5 Traveling 2 17.9652778 February
## 7 3 1 2 Traveling 1 0.9509375 January
## 8 3 1 4 Traveling 1 2.7434028 January
## 9 3 1 6 Traveling 2 2.4634375 February
## 10 4 1 2 Traveling 1 4.4351968 January
## 11 4 1 4 Traveling 1 5.9662500 January
## 12 4 1 6 Traveling 2 11.4628819 February## Source: local data frame [19 x 7]
## Groups: Animal, Track, Bout, phistate [19]
##
## Animal Track Bout phistate MonthF Days Month
## (dbl) (dbl) (int) (fctr) (dbl) (dbl) (fctr)
## 1 1 1 1 Area-restricted Search 2 8.8687500 February
## 2 1 1 3 Area-restricted Search 2 29.7638889 February
## 3 1 1 5 Area-restricted Search 3 22.4395833 March
## 4 1 1 7 Area-restricted Search 4 8.8083333 April
## 5 2 1 2 Area-restricted Search 2 4.8215278 February
## 6 2 1 4 Area-restricted Search 2 1.8284722 February
## 7 2 1 6 Area-restricted Search 3 3.4305556 March
## 8 3 1 1 Area-restricted Search 1 9.9591435 January
## 9 3 1 3 Area-restricted Search 1 6.4850694 January
## 10 3 1 5 Area-restricted Search 2 10.9568750 February
## 11 3 1 7 Area-restricted Search 2 2.3541204 February
## 12 4 1 1 Area-restricted Search 1 5.9826852 January
## 13 4 1 3 Area-restricted Search 1 2.9365278 January
## 14 4 1 5 Area-restricted Search 1 9.9229398 January
## 15 4 1 7 Area-restricted Search 2 5.0989120 February
## 16 4 2 1 Area-restricted Search 2 18.9131829 February
## 17 4 3 1 Area-restricted Search 3 0.6460301 March
## 18 4 4 1 Area-restricted Search 3 1.3696991 March
## 19 4 5 1 Area-restricted Search 3 1.0939931 March## Source: local data frame [31 x 7]
## Groups: Animal, Track, Bout, phistate [31]
##
## Animal Track Bout phistate MonthF Days Month
## (dbl) (dbl) (int) (fctr) (dbl) (dbl) (fctr)
## 1 1 1 1 Area-restricted Search 2 8.868750 February
## 2 1 1 2 Traveling 2 1.351389 February
## 3 1 1 3 Area-restricted Search 2 29.763889 February
## 4 1 1 4 Traveling 3 2.471528 March
## 5 1 1 5 Area-restricted Search 3 22.439583 March
## 6 1 1 6 Traveling 4 1.475000 April
## 7 1 1 7 Area-restricted Search 4 8.808333 April
## 8 2 1 1 Traveling 2 6.340278 February
## 9 2 1 2 Area-restricted Search 2 4.821528 February
## 10 2 1 3 Traveling 2 2.800000 February
## .. ... ... ... ... ... ... ...
5.1 Proportion of time allocation
## Month Traveling Area-restricted Search PropF TotalTime
## 1 January 14.095787 35.286366 0.7145571 49.38215
## 2 February 42.383264 82.605729 0.6609040 124.98899
## 3 March 2.471528 28.979861 0.9214175 31.45139
## 4 April 1.475000 8.808333 0.8565640 10.283335.2 Number of bouts
6 Time spent in grid cell
6.1 All years
6.2 Add in sea ice
